Sampling Theory, a Renaissance: Compressive Sensing and Other Developments (Applied and Numerical Harmonic Analysis)
Reconstructing or approximating objects from seemingly incomplete information is a frequent challenge in mathematics, science, and engineering. A multitude of tools designed to recover hidden information are based on Shannon’s classical sampling theorem, a central pillar of Sampling Theory. The growing need to efficiently obtain precise and tailored digital representations of complex objects and phenomena requires the maturation of available tools in Sampling Theory as well as the development of complementary, novel mathematical theories. Today, research themes such as Compressed Sensing and Frame Theory re-energize the broad area of Sampling Theory. This volume illustrates the renaissance that the area of Sampling Theory is currently experiencing. It touches upon trendsetting areas such as Compressed Sensing, Finite Frames, Parametric Partial Differential Equations, Quantization, Finite Rate of Innovation, System Theory, as well as sampling in Geometry and Algebraic Topology.
Why Read This Book
You will get a contemporary, mathematically grounded tour of sampling theory that connects Shannon’s classical results to modern advances such as compressed sensing and frame theory. You will learn how these ideas reshape practical signal acquisition and reconstruction across audio, radar, and communications, giving you both conceptual clarity and pointers to useful algorithms.
Who Will Benefit
Graduate students, researchers, and senior engineers in signal processing, applied mathematics, and communications who want to bridge classical sampling theory with modern sparse and frame-based recovery methods.
Level: Advanced — Prerequisites: Undergraduate-level linear algebra and calculus, familiarity with basic probability and Fourier analysis, and prior exposure to digital signal processing concepts (e.g., sampling, filters, FFT).
Key Takeaways
- Understand the theoretical foundations that generalize Shannon sampling — including frames, irregular sampling, and finite rate of innovation models.
- Apply compressed sensing principles to formulate and analyze sparse recovery problems relevant to audio, radar, and communications.
- Analyze stability, noise sensitivity, and robustness of sampling and reconstruction schemes using spectral and statistical tools.
- Design and evaluate multiresolution and wavelet-based sampling strategies and their connections to filter banks and FFT-based methods.
- Employ convex optimization and algorithmic frameworks (e.g., L1 minimization, greedy algorithms) for practical reconstruction tasks.
- Identify open problems and translate advanced sampling ideas into domain-specific systems such as adaptive filters and spectral estimation pipelines.
Topics Covered
- 1. Introduction: The Renaissance of Sampling Theory
- 2. Classical Shannon Sampling and Its Limits
- 3. Frames, Irregular Sampling, and Redundancy
- 4. Finite Rate of Innovation and Nonbandlimited Models
- 5. Fundamentals of Compressed Sensing and Sparse Models
- 6. Algorithms for Sparse Recovery: Convex and Greedy Methods
- 7. Wavelets, Multiresolution Analysis, and Filter Banks
- 8. Spectral Analysis, FFT Methods, and Numerical Aspects
- 9. Statistical Signal Processing: Noise, Estimation, and Robustness
- 10. Applications: Audio and Speech Processing
- 11. Applications: Radar and Remote Sensing
- 12. Applications: Communications and Sampling in Networks
- 13. Implementation Considerations and Software Tools
- 14. Open Problems and Future Directions in Sampling Theory
Languages, Platforms & Tools
How It Compares
Complements Foucart & Rauhut's Compressed Sensing (which focuses tightly on CS theory and proofs) and Mallat's A Wavelet Tour (which emphasizes wavelets and applications); this volume synthesizes sampling, frames and CS across theory and applied examples.
